Mirror Laboratory
Project «Quaternions, geometric algebras, and applications»
The project 'Quaternions, geometric algebras, and applications' of the Laboratory for Geometric Algebra and Applications in collaboration with the international research laboratory 'Multiscale Mathematical Modeling and Computer Calculations' of the Federal State Autonomous Educational Institution of Higher Education 'M. K. Ammosov North-Eastern Federal University' (NEFU, Yakutsk), has won the competition for the creation of Mirror Laboratories 2024.
The aim of the project is to develop new algebraic, geometric, and computational methods related to geometric algebras and quaternions and to apply these methods in image processing, computer science, and physics.
Implementation period: 2024 – 2026.
Participants from HSE University
Project Head; Doctor of Physical and Mathematical Sciences; Head of Laboratory for Geometric Algebra and Applications; Professor at the Department of Mathematics, Faculty of Economic Sciences
Doctor of Physical and Mathematical Sciences; Leading Research Fellow, Senior Research Fellow at Laboratory for Geometric Algebra and Applications; Leading Research Fellow at the Steklov Mathematical Institute of the Russian Academy of Sciences
Research Assistant at Laboratory for Geometric Algebra and Applications; Doctoral Student at Doctoral School of Computer Science, 1st year
Research Assistant at Laboratory for Geometric Algebra and Applications; Assistant, Doctoral Student at the School of Applied Mathematics, 3rd year, MIEM HSE
Research Assistant at Laboratory for Geometric Algebra and Applications; Doctoral Student at Doctoral School of Computer Science, 1st year; Lecturer at the Department of Mathematics, Faculty of Economic Sciences
Doctoral Student at Doctoral School of Computer Science, 1st year
Master's Student at Faculty of Computer Science, HSE University
Participants from NEFU
Project Head; Doctor of Physical and Mathematical Sciences, Professor; Head of the Department of 'Computational Technologies' at NEFU
Professor, Linyi University
PhD, NEFU
Postgraduate Student, NEFU
Postgraduate Student, NEFU
School for Students and Young Scientists 'Quaternions, Geometric Algebras, and Applications'
School for Students and Young Scientists 'Quaternions, Geometric Algebras, and Applications' is organized by the Laboratory for Geometric Algebra and Applications (HSE University) in collaboration with the international research laboratory 'Multiscale Mathematical Modeling and Computer Calculations' of the Federal State Autonomous Educational Institution of Higher Education 'M. K. Ammosov North-Eastern Federal University' (NEFU, Yakutsk) as part of the project 'Mirror Laboratories (HSE Univeristy): 'Quaternions, Geometric Algebras, and Applications'.
1) Dates: November 15–17, 2024.
Format: Online.
Website: https://economics.hse.ru/en/gaaschool2024/
2) Dates: November 14–16, 2025.
Format: Online.
Publications on the project
- Shirokov D. On Rank of Multivectors in Geometric Algebras // Mathematical Methods in the Applied Sciences. – 2025. – Vol. 48, No. 11. – P. 11095-11102. – DOI: 10.1002/mma.10946 (List A)
- Shirokov D. Calculation of Spin Group Elements Revisited // International Journal of Geometric Methods in Modern Physics. – 2025. – DOI: 10.1142/S0219887825400316 (List C)
- Марчук Н.Г. Классификация полевых уравнений для спиноров Вейля и для Elko спиноров // Дифференциальные уравнения. – 2025. – Т. 61, № 3. – С. 366-373. – DOI: 10.7868/S3034503025030069 (List C)
- Zhang D., Jiang T., Vasil’ev V.I., Wang G. Two efficient algorithms for the commutative quaternion equality constrained least squares problem // Journal of Computational and Applied Mathematics. – 2026. – Vol. 474. – P. 116964. – DOI: 10.1016/j.cam.2025.116964 (List A)
- Guo Z., Jiang T., Wang G., Vasil’ev V.I. An efficient algorithm for the eigenvalue problem of a Hermitian quaternion matrix in quantum chemistry // Journal of Computational and Applied Mathematics. – 2025. – Vol. 463. – P. 116516. – DOI: 10.1016/j.cam.2025.116516 (List A)
- Filimoshina E., Shirokov D. GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras // Volume 267: International Conference on Machine Learning, 13-19 July 2025, Vancouver Convention Center, Vancouver, Canada. – 2025. – P. 17153-17188. – https://proceedings.mlr.press/v267/filimoshina25a.html (List A)
- Sharma H., Shirokov D. Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras // Advances in Applied Clifford Algebras. – 2025. – Vol. 35 – P. 44. – DOI: 10.1007/s00006-025-01406-6 (List C)
- Shirokov D. On Unitary Groups in Ternary and Generalized Clifford Algebras // Advances in Applied Clifford Algebras. – 2025. – Vol. 35. – P. 25. – DOI: 10.1007/s00006-025-01388-5 (List C)
- Wang G. An unstructured algorithm for the singular value decomposition of biquaternion matrices // Applied Mathematics Letters. – 2025. – Vol. 163. – P. 109436. – DOI: 10.1016/j.aml.2024.109436 (List A)
- Sharma H., Shirokov D.S. On Commutative Analogues of Clifford Algebras and Their Decompositions // Advances in Applied Clifford Algebras. – 2026. – DOI: 10.1007/s00006-025-01422-6 (in print) (List C)
- Wang G. An efficient complex structure-preserving algorithm for the Autonne-Takagi decomposition of quaternion matrices // Under review
- Zhang D. A new color image watermarking scheme based on Schur decomposition of commutative quaternion matrices // Under review
- Guo Z. An efficient complex structure-preserving algorithm for Schur decomposition of quaternion matrix // Under review
Scientific seminar 'Quaternions, geometric algebras, and applications'
The regular scientific seminar is held online via the Zoom link.
Co-organizers of the seminar: Dmitry Shirokov (HSE University, dm.shirokov@gmail.com) and Vasily Vasil'ev (NEFU).
Secretary of the seminar: Ekaterina Filimoshina (efilimoshina@hse.ru).
We invite all interested students, graduate students, and researchers to participate in the seminar as listeners or speakers. To receive notifications about upcoming seminar sessions, please contact the seminar secretary (efilimoshina@hse.ru).
-
16. On the Possibility of Representing Clifford Algebras and Lie Algebras Using Graphs
Date: 13.12.2025, 14:00
Speaker: Angelina Iudina (HSE University)
Abstract:
Establishing the relationship between Clifford algebras and Lie algebras with graphs is a nontrivial mathematical problem. Its solution can significantly expand the tools for studying the internal structure of these algebras through the use of classical graph analysis methods. This talk examines approaches to constructing Clifford quasi-algebras with a one-dimensional center by combining graphs associated with the original algebraic structures, as well as methods for constructing Lie algebras from graphs and methods for their subsequent decomposition and classification.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1110451890.html
Video of the seminar: https://vk.com/video-227370571_456239056
-
15. Lanczos Equation (1929) in the Modified Standard Model of Elementary Particles
Date: 04.12.2025, 11:00
Speaker: Nikolay Marchuk (Steklov Mathematical Institute of the Russian Academy of Sciences and HSE University)
Abstract:
This talk discusses the interim results of a project to modify the Standard Model of elementary particles and their interactions, based on the Lanczos equation (1929) to describe leptons. The report is based on two papers by the author, published in the journal Theoretical and Mathematical Physics in 2024 and in the journal Differential Equations in 2025.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1107800786.html
Video of the seminar: https://vk.com/video-227370571_456239055
-
14. On Bilinear Covariants and Fierz Identities
Date: 06.11.2025, 10:00
Speaker: Sofiia Rumiantseva (HSE University)
Abstract:
This talk examines bilinear covariants and Fierz identities arising in the analysis of solutions to the Dirac equation and its real analog, the Dirac–Hestenes equation, in Minkowski space of signature (1,3). Constructions of the Fierz aggregate and 'boomerangs' that allow one to reconstruct a spinor from its bilinear forms are discussed. The possibility of generalizing these relations to the multidimensional case for the Dirac–Hestenes equation is also analyzed.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1104867021.html
Video of the seminar: https://vk.com/video-227370571_456239042
-
13. Clifford Equivariant Denoising Diffusion Models for Molecular Generation
Date: 09.10.2025, 10:00
Speaker: Sergey Kudriashov (HSE University)
Abstract:
Today, diffusion models have become widely used in generative problems. However, the probabilistic nature of these models limits their applicability to problems requiring structure preservation, such as molecular generation, forcing researchers to resort to data augmentation and other heuristic methods. A recent paper by Liu et al. (2025) (https://arxiv.org/pdf/2504.15773) proposed using a Clifford algebra framework to obtain architecturally equivariant models. This talk will include a brief introduction to the framework of diffusion models in discrete and continuous time, as well as an overview of the aforementioned work.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1104852842.html
-
12. Biquaternions and Second-order Matrices in Field Theory Equations
Date: 11.09.2025, 10:00
Speaker: Nikolay Marchuk (Steklov Mathematical Institute of the Russian Academy of Sciences and HSE University)
Abstract:
We will discuss biquaternions and second-order matrices as the main mathematical structures of the conservative model of field theory.
Photo of the seminar: https://economics.hse.ru/gaa/news/1104853492.html
Video of the seminar: https://vk.com/video-227370571_456239041
-
11. GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras
Date: 02.08.2025, 10:00
Speaker: Ekaterina Filimoshina (HSE University)
Abstract:
We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters. This talk is based on the work presented at ICML 2025.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1073390979.html
Video of the seminar: https://vk.com/video-227370571_456239040
-
10. On the Application of Clifford Algebras to Word Embeddings Construction
Date: 19.04.2025, 10:00
Speaker: Vladislav Kopchev (HSE University)
Abstract:
The talk discusses modifications of classical approaches to constructing word embeddings that allow moving from word representations as vectors to representations as multivectors from Clifford algebras. It is shown that approaches based on Clifford algebras allow obtaining higher metrics on classical datasets and broader opportunities for interpreting models.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1037889450.html
Video of the seminar: https://vk.com/video-227370571_456239039
-
9. On the Inverse Element Formula in Clifford Algebras in the Case n≤7
Date: 21.02.2025, 10:00
Speaker: Kamron Abdulkhaev (HSE University)
Abstract:
The talk is devoted to the analysis of the method proposed by Eckhard Hitzer and Stephen J. Sangwine in the article "Construction of multivector inverse for Clifford algebras over 2m+1-dimensional vector spaces from multivector inverse for Clifford algebras over 2m-dimensional vector spaces". In this approach, the results for even dimensions (2m) are used to construct inverse elements in odd cases (2m+1). The report considers the application of this method in the case n≤7, as well as its efficiency in deriving generalized basis-free formulas.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1022903032.html
Video of the seminar: https://vk.com/video-227370571_456239038
-
8. Special Basis in Commutative Analogues of Clifford Algebras and their Irreducible Real Representations
Date: 01.02.2025, 10:00
Speaker: Heerak Sharma (Indian Institute of Science Education and Research (IISER), Pune, India)
Abstract:
In this talk, we will first discuss isomorphism classes in commutative analogues of Clifford algebras K_{p,q}s. In Clifford algebra we have the Cartan—Bott theorem which says that Clifford algebras have 5 different isomorphism classes depending on the value of p-q (mod8). We will see that a simpler result holds for K_{p,q}s: a K_{p,q} is isomorphic to either K_{p+q,0} or K_{0,p+q} depending on whether p > 0 or not. Next we will explore K_{n,0} and K_{0,n} and see that they are isomorphic to a direct sum of a bunch of real numbers or a bunch of complex numbers respectively. We will give explicit basis in K_{n,0} and K_{0,n} that make their direct sum decomposition apparent. Lastly, as an application of the results we develop in the talk, we will work out all irreducible real representations of K_{p,q}s and see that they come from irreducible complex representations of K_{p,q} as conjectured in my last talk.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1022893018.html
Video of the seminar: https://vk.com/video-227370571_456239037
-
7. Invariance of the Multidimensional Dirac—Hestenes Equation under Lorentz Transformations
Date: 22.01.2025, 11:00
Speaker: Sofiia Rumiantseva (HSE University)
Abstract:
The talk considers the invariance of the multidimensional Dirac—Hestenes equation under Lorentz transformations, which are orthogonal transformations of coordinates. The multidimensional Dirac—Hestenes equation is considered in the subalgebra of the real Clifford algebra Cl(1,n). The talk provides a detailed analysis of the classical Dirac—Hestenes equation for the signature (1,3), and then generalizes the obtained results to the multidimensional case for the signature (1,n).
Photo of the seminar: https://economics.hse.ru/en/gaa/news/1022892522.html
Video of the seminar: https://vk.com/video-227370571_456239036
-
6. More Representations of Commutative Analogues of Clifford Algebras and Notion of Eigenvalues in them
Date: 05.12.2024, 10:00
Speaker: Heerak Sharma (Indian Institute of Science Education and Research (IISER), Pune, India)
Abstract:
In my last few talks we had discussed a representation for K_{p,q}s. In this talk, we will discuss a few more representations of K_{p,q}. We will start by showing that the representation we knew for K_{p,q}s is special: it is the regular representation of the algebra K_{p,q}. Next, we will find all complex irreducible representations of K_{p,q}. To do this, we will use concepts from character theory (of representations) and group algebras. We will also look at some real irreducible representations. Next, we will see a nice factorisation for the determinant of an element of K_{p,q} introduced in my last talk. Lastly, we will define a notion of eigenvalue in K_{p,q} and show that the eigenvalues are exactly the complex irreducible representations.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/989234733.html
Video of the seminar: https://vk.com/video-227370571_456239035
-
5. On Trace, Determinant, Other Coefficients of Characteristic Polynomials in Commutative Analogues of Clifford Algebras
Date: 09.11.2024, 10:00
Speaker: Heerak Sharma (Indian Institute of Science Education and Research (IISER), Pune, India)
Abstract:
In my last talk we had discussed a matrix representation of the algebras K_{p,q}. In this talk, we will use the representation of K_{p,q} to associate notions of trace, determinant and characteristic polynomial with the elements of K_{p,q}. We will then give explicit expressions, not involving the matrix representations used to define them, for the trace, determinant and coefficients in characteristic polynomial. Lastly, as an application of formula for determinant, we will give formulas for inverses of invertible elements in K_{p,q}s and show that non-invertible elements in K_{p,q} are zero divisors.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/984824970.html
Video of the seminar: https://vk.com/video-227370571_456239023
-
4. Development of Quaternion Matrix Decomposition Algorithms and Their Applications
Date: 17.10.2024, 10:00
Speaker: Zhang Dong (NEFU)
Abstract:
In this talk, we will discuss the research that formed the basis of my PhD thesis, including the development of fast algorithms for singular value decomposition of quaternion matrices, as well as commutative quaternion matrices decomposition and least squares problems, with subsequent integration of these algorithms into signal and color image processing problems.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/976316210.html
Video of the seminar: https://vk.com/video-227370571_456239021
-
3. On Multidimensional Dirac—Hestenes Equation in Geometric Algebras
Date: 03.10.2024, 10:00
Speaker: Sofiia Rumiantseva (HSE University)
Abstract:
We investigate the multidimensional Dirac–Hestenes equation in the formalism of the real geometric algebra Cl(1,n). It is well known that the classical four-dimensional Dirac equation is equivalent to the Dirac–Hestenes equation in the geometric algebra Cl(1,3). It means that solutions of one equation can be obtained from solutions of the other and vice versa. A significant advantage of the Dirac–Hestenes formulation is that its wave function is entirely real, providing a deeper understanding of geometric aspects of problems. This work extends the theory to higher dimensions. Due to the fact that the matrix representation of the complexified geometric algebra depends on the parity of n, the cases of even and odd n are analyzed separately. In the even case, two types of spinors, semi-spinors and double spinors, which are solutions to the Dirac equation, are investigated.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/969848113.html
Video of the seminar: https://vk.com/video-227370571_456239019
-
2. Algorithms for Computing Eigenvalues of Quaternion Matrices and Their Applications
Date: 26.09.2024, 10:00
Speaker: Guo Zhenwei (NEFU)
Abstract:
In the talk, we introduce new algorithms for solving the eigenvalue problem for three classes of quaternion matrices, including quaternions, split-quaternions, and commutative quaternions, with applications to solving problems in color image processing.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/966692197.html
Video of the seminar: https://vk.com/video-227370571_456239018
-
1. Class of Field Equations for Neutrinos and Other Particles with Spin 1/2
Date: 12.09.2024, 10:00
Speaker: Nikolay Marchuk (Steklov Mathematical Institute of the Russian Academy of Sciences and HSE University)
Abstract:
A new equation (class of equations) is introduced, which is considered as a candidate for an equation for neutrinos with non-zero mass.
Photo of the seminar: https://economics.hse.ru/en/gaa/news/961677612.html
Video of the seminar: https://vk.com/video-227370571_456239017
Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!
To be used only for spelling or punctuation mistakes.